Contention issues in congestion games Elias Koutsoupias Katia - - PowerPoint PPT Presentation

Contention issues in congestion games

Elias Koutsoupias Katia Papakonstantinopoulou

University of Athens

ICALP – Warwick, July 2012

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 1 / 18

Motivation - Problem Description

Motivation

Games in which players can time their participation with the hope that fewer players will compete for the same resources.

TCP congestion control policy is such a strategy

A first step to the study of the important class of congestion games with time-dependent strategies.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 2 / 18

Motivation - Problem Description

Congestion and Contention

Congestion Contention as in Internet routing as in Ethernet / wireless protocols

Stations A B C D Time

resource sharing ⇒ higher cost resource sharing ⇒ nobody succeeds Strategy: Set of resources Strategy: Timing In between: The cost depends on both the set of selected resources and timing (eg. TCP). ⇓ Strategy: Set of resources + Timing

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 3 / 18

Motivation - Problem Description

Our game-theoretic abstraction

Congestion game with time dimension Strategy: which path to use and when (probability pe,t) Payoff: depends on the number of users using the same links at the same time Assumptions: underlying network: A set of parallel links with affine latencies.

link e, k users

. . .

cost for each user: ℓe(k) = aek + be

strategies: non-adaptive, symmetric

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 4 / 18

Outline

1

Related Work

2

Our Work latency models & derived games structural properties of conveyor belt games study of symmetric Nash equilibria in boat model study of symmetric Nash equilibria in conveyor belt model

3

Open Problems

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 5 / 18

Related Work

Related Work

Strategy: Timing Time dependent strategies Classical Congestion Games

?

Games with (Strategy: Set of Resources) Games with for Contention resolution time-dependent costs We work here Game theoretic Models Game theoretic internet routing Strategy: Transmission rate, etc Atomic Non Atomic

PoA [KP] PoS [Ansh+] [Ros73] PoA [RT] [Bhaskar+] [Ansh+] [Fiat+] [Christod+] Packet switching TCP-like games [Kessel+] [Akella+]

• f selfish behavior

Contention Resolution

[MacKenzie+] slotted ALOHA [Altman+]

Games

[Koch+] [Macko+] [Hoefer+] Congestion Control [Garg+] [Altman+]

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 6 / 18

Our Work latency models & derived games

Boat model

the latency of a player is influenced only by the players that start at the same time for each link:

1 2 t

. . .

time cost = t + original congestion cost The speed of each boat depends only on the number of players on it.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 7 / 18

Our Work latency models & derived games

Conveyor belt model

the latency of a player is affected by the players that share the system, even if they started earlier or later

t1 t2 f1 f2 time

• n each link:

t2−t1 ℓ(1) + f1−t2 ℓ(2) = 1 unit of work (distance)

• The speed depends on the number of people on the belt:

During each time step, if k players use this link, each one completes work of

1 ℓ(k).

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 8 / 18

Our Work summary of results

Our results at a glance

Boat Conveyor belt congestion game

• nly for 2 players

existence of pure NE

• not always

exact network topology matters? No Yes nature of unique, symmetric NE probabilities drop linearly with time nature of optimal symmetric solution structure that resembles the NE PoA, PoS small (1.06)

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 9 / 18

Our Work structural properties of conveyor belt games

Are the conveyor belt games congestion games?

Only 2 player conveyor belt games are congestion games! For 2 players and arbitrary networks, there is a potential function.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 10 / 18

Our Work structural properties of conveyor belt games

Are the conveyor belt games congestion games?

Only 2 player conveyor belt games are congestion games! For 2 players and arbitrary networks, there is a potential function. For 3 or more players (even on a single link): There are games that have no pure (asymmetric) equilibria, so they are not in general congestion games.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 10 / 18

Our Work structural properties of conveyor belt games

Conveyor belt games may not possess pure NE

No pure NE for 3 players on one link with ℓ(k) = 5k − 1!

1 link time 3 players

t1 f1 t2 f2 t3 f3

We assume that they overlap. (The other case is similar.)

t2−t1 ℓ(1) + t3−t2 ℓ(2) + f1−t3 ℓ(3) = 1

• & similar equations

for the other 2 players

Best strategy for player 3: select t3 ≥ f1 (no overlap - contradiction).

⇓

There are not finish times that satisfy this game’s constraints.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 11 / 18

Our Work structural properties of conveyor belt games

In conveyor belt games the network topology matters

In conveyor belt games, a user’s cost depends on the underlying network topology.

k k + 1 k + 1 k k + 1 k + 1

reverse

Consider 2 players. They finish at f1 = 7/2, f2 = 9/2. On the reversed they finish at .

⇓

The finish time of each player is not the same in these two networks!

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 12 / 18

Our Work structural properties of conveyor belt games

In conveyor belt games the network topology matters

In conveyor belt games, a user’s cost depends on the underlying network topology.

k k + 1 k + 1 k k + 1 k + 1

reverse

Consider 2 players. They finish at f1 = 7/2, f2 = 9/2. On the reversed they finish at f ′

1 = 4 > f1, f ′ 2 = 5 > f2.

⇓

The finish time of each player is not the same in these two networks!

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 13 / 18

Our Work study of symmetric Nash equilibria in boat model

The structure of symmetric mixed Nash Equilibria and

• ptimal non-selfish solution in boat model

Both of them: are unique in each link the probabilities drop linearly with time

effect of selfishness

Nash equilibrium Optimal Probability Time 1 link (a=1, b=0) 100 players

◮ We observe the bicriteria relation (also in [RT02]). ◮ Users are more greedy in NE than in OPT in the beginning of the game.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 14 / 18

Our Work study of symmetric Nash equilibria in boat model

PoA/PoS of symmetric strategies in boat model

For a fixed network, the PoA tends to 3 √ 2/4 ≈ 1.06 (assuming that number of players → ∞) for a fixed number of players:

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 15 / 18

Our Work study of symmetric Nash equilibria in conveyor belt model

Structure of symmetric mixed Nash Equilibria and

• ptimal non-selfish solution in conv. belt model (2 players)

Very similar to boat model, BUT: the probabilities here are non-zero only at multiples of ℓ(1). ⇒ Either do not overlap or start together!

Probability Time Nash Equilibrium Optimal 1 link latencies: ℓ(1) = 3 ℓ(2) = 19

effect of selfishness

◮ Bicriteria relation. ◮ For a fixed network, PoA tends to 3 √ 2/4 ≈ 1.06 (assuming latency → ∞)

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 16 / 18

Open Problems

Open problems - ongoing work

Essentially any issue not covered in this talk is open!

More general configurations More complicated Other variants of the problem

more players (conv. belt) more general latency functions (boat) more general networks adaptive strategies preemption jobs coming online (no waiting time)

• nly the past influences the delay in each link (conv. belt)

players with weights non-adaptive strategies parallel links boat model conveyor belt model

strategies

non atomic games

Ongoing work: general networks adaptive strategies.

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 17 / 18

thank you for your attention

Katia Papakonstantinopoulou (U Athens) Contention issues in congestion games ICALP 2012 18 / 18